Untangling polygons and graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.Comment: 11 pages, 3 figure

β\beta-Stars or On Extending a Drawing of a Connected Subgraph

We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing $\Gamma_H$ of an induced connected subgraph $H$ can be extended with at most $\min\{ h/2, \beta + \log_2(h) + 1\}$ bends per edge, where $\beta$ is the largest star complexity of a face of $\Gamma_H$ and $h$ is the size of the largest face of $H$. This result significantly improves the previously known upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

Planar Drawings of Fixed-Mobile Bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

A polynomial-time algorithm to design push plans for sensorless parts sorting

We consider the efficient computation of sequences of push actions that simultaneously orient two different polygons. Our motivation for studying this problem comes from the observation that appropriately oriented parts admit simple sensorless sorting. We study the sorting of two polygonal parts by first putting them in properly selected orientations. We give an O(n2 log n)-time algorithm to enumerate all pairs of orientations for the two parts that can be realized by a sequence of push actions and admit sensorless sorting. We then propose an O(n4 log2 n)-time algorithm for finding the shortest sequence of push actions establishing a given realizable pair of orientations for the two parts. These results generalize to the sorting of k polygonal parts

A polynomial-time algorithm to design push plans for sensorless parts sorting

We consider the efficient computation of sequences of push actions that simultaneously orient two different polygons. Our motivation for studying this problem comes from the observation that appropriately oriented parts admit simple sensorless sorting. We study the sorting of two polygonal parts by first putting them in properly selected orientations. We give an O(n2 log n)-time algorithm to enumerate all pairs of orientations for the two parts that can be realized by a sequence of push actions and admit sensorless sorting. We then propose an O(n4 log2 n)-time algorithm for finding the shortest sequence of push actions establishing a given realizable pair of orientations for the two parts. These results generalize to the sorting of k polygonal parts

Disjoint unit spheres admit at most two line transversals

We show that a set of n disjoint unit spheres in R d admits at most two distinct geometric permutations, or line transversals, if n is large enough. This bound is optimal

Untangling a planar graph

A straight-line drawing d of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,d) denote the minimum number of vertices that need to be moved to untangle d. We show that shift(G,d) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem. Further we define fix(G,d)=n-shift(G,d) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling d. We give an algorithm that fixes at least vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing d G of G with and an n-vertex outerplanar graph H and a drawing d H of H with . Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs

Moving vertices to make drawings plane

In John Tantalo’s on-line game Planarity the player is given a non-plane straight-line drawing of a planar graph. The aim is to make the drawing plane as quickly as possible by moving vertices. In this paper we investigate the related problem MinMovedVertices which asks for the minimum number of vertex moves. First, we show that MinMovedVertices is NP-hard and hard to approximate. Second, we establish a connection to the graph-drawing problem 1BendPointSetEmbeddability, which yields similar results for that problem. Third, we give bounds for the behavior of MinMovedVertices on trees and general planar graphs. This work was started on the 9th "Korean" Workshop on Computational Geometry and Geometric Networks organized by A. Wolff and X. Goaoc, July 30–August 4, 2006 in Schloß Dagstuhl, Germany. Further contributions were made at the 2nd Workshop on Graph Drawing and Computational Geometry organized by W. Didimo and G. Liotta, March 11–16, 2007 in Bertinoro, Italy